Finite-dimensional weak and norm topologies coincide
= Finite-dimensional weak and norm topologies coincide
On a <finite-dimensional vector space> equipped with a <norm>, the <weak topology> and <norm topology> coincide. The weak topology is no finer because every element of the dual is norm-continuous. Conversely, finitely many coordinate functionals in a basis control the norm, so every sufficiently small basic weak neighbourhood lies in a prescribed norm ball.